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analysis question

Post Icon Posted: Submitted by richardhp on 12 July 2008 - 10:35am.

Joined: 2007-10-01
Posts: 179

I was wondering whether the sequence $ log^n(n) $ would converge or not. obviously $ log^k(n) $ diverges for any fixed k, but i thought maybe the other one didn't.

‹ Books 1000 posts! ›
Post Icon Posted: 12 July 2008 - 3:38pm

Joined: 2007-10-03
Posts: 397

What do you mean by $ \log^k(n) $ ?

Do you mean $ \underbrace{\log(\log(\log(\cdots(n)\cdots)))}_{k \text{ logs}} $ ?

Because, if so, $ \log^n(n) $ fails to be defined past $ n=3 $ if you're working in the reals.

In the complex numbers, it converges to the only solution to $ \log(x) = \log(\log(x)) $ (or $ x = \log(x) $), which is approximately $ 0.318131505205+1.33723570143i $; my CAS solves it as $ e^{-\omega(\pi i)} $, using the Wright Omega function which I have never heard of.

If you mean $ \left ( \log(n) \right )^k $ then it obviously diverges to $ + \infty $ (and you would fail at life for using that notation).

Edit : I created a Wikipedia article on the function, see http://en.wikipedia.org/wiki/Wright_Omega_Function
Post Icon Posted: 12 July 2008 - 9:19pm

Joined: 2006-11-02
Posts: 1090

and you would fail at life for using that notation

There's worse: $ \ln $ instead of $ \log $ (which obviously warranted a correction :P).
Post Icon Posted: 13 July 2008 - 6:55pm

Joined: 2007-10-03
Posts: 397

Okay, own up : who sneaked the toaster allusions into my first Wikipedia article and why ?
Post Icon Posted: 13 July 2008 - 7:39pm

Joined: 2008-06-21
Posts: 27

Right, it was me.

Post Icon Posted: 14 July 2008 - 8:57am

Joined: 2008-06-14
Posts: 3

no, it was me.

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